The Federal Bureau of Investigation is warning U.S. banks and online businesses about hackers in Eastern Europe breaking into computers, stealing credit-card information on hundreds of thousands of customers, and then issuing blackmail threats, some of which exceed $100,000.

The agency said more than 40 businesses in 20 states have been hit by hacker rings working together in Russia and the Ukraine, and that more than a million credit-card numbers have been stolen. FBI officials said hundreds more companies may have been...